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I've used an HP-28C and 15C for years. Two weeks ago my sons gave me an HP-42S with both Owner's Manual and Programming Manual. I have figured everything out for the most part except for roots of polynomials. I can figure quadratics and cubics such as x^3+3x^2+9x-1 but I can't get the programming syntax for 3x^3-9x^2+14x+7 nor can I figure any kind of quadratics. Please help me. These are a breeze to figure on the 15C and 28C. Also, is there any kind of program for the 42S that can compute complex roots of polynomials?

Hi.

I guess the closest technique is the Horner's method. Think of

3x^3-9x^2+14x+7

((3*x-9)*X+14)*X+7

LBL "XMPL"

MVAR "X"

RCL "X"

ENTER

ENTER

ENTER

3

*

9

-

*

14

+

*

7

+

END

[SHIFT][SOLVER]

[X=-0.3897]

Hope it helps.

Cheers.

Luiz (Brazil)

Edited: 28 June 2012, 1:12 p.m. after one or more responses were posted

There is no need for Horner's method with this one, you can simply, though unelegantly key in

LBL TST

MVAR "X"

RCL "X"

3

Y^X

3

*

RCL "X"

X^2

9

*

-

RCL "X"

14

*

+

7

+

RTN

...and than solve TST for X, that yields X=-0.3897

As this is pretty straightforward, maybe I have not completely understood what the initial problem with this is...?

Edited: 28 June 2012, 4:40 a.m.

Hi.

I just considered that by using Horner's method to reduce a polynomial expression to a few arithmetic operations with no need for powers/exponents, the final program would be easier to be written. 8^)

Cheers.

Luiz (Brazil)

Right, in this case two program step are spared :

001 LBL "TST        001 LBL "XMP

002 MVAR "X"        002 MVAR "X"

003 RCL "X"         003 RCL "X" 

004 3               004 ENTER   

005 Y^X             005 ENTER   

006 3               006 ENTER   

007 *               007 3       

008 RCL "X"         008 *       

009 X^2             009 9       

010 9               010 -       

011 *               011 *       

012 -               012 14      

013 RCL "X"         013 +       

014 14              014 *       

015 *               015 7       

016 +               016 +       

017 7               017 RTN     

018 +                           

019 RTN                         

Very similar in fact

Thanks so much for helping me with the cubic equation. I tried both methods given to me to find the roots for the following quartics and I get only one root for each equation:

x^4+8x^3+9x^2-8x-10=0
4x^4-31x^2+21x+18=0

I am obviously doing something wrong.

I only tried the first one but could find all four roots: -6.4495, -1.5505, -1 and 1.

It all depends on the right initial guesses, use the programmed function to calculate some values to plot it roughly, then you put in two guesses for x that are located in the vicinity of the potential roots. You do this by starting the solver, selecting the programmed function and than keying in the lower guess, press [X], upper guess, press [X] and than [X] again to solve. See the manual on further details.

Edited: 28 June 2012, 5:08 p.m. after one or more responses were posted

What program syntax did you use to solve the first equation? I will copy and save it for future reference.

That’s the problem with numeric solver; you have to initiate each root search with an appropriate initial value.
There is no trick to find all these values by only looking at the polynomial. That is why graphic calculators, such as the HP-28C, will spare a lot of time. It is a good advice to start plotting the curve.

For x^4+8x^3+9x^2-8x-10, enter the polynomial equation into you HP-28C :

 « X 8 + X * 9 + X * 8 – X * 10 - »  STEQ

(10,31) DUP PMAX CHS PMIN

DRAW

You may obtain the following graph, using cursor keys and INS key you simply have to plot positions close to the four roots. Then press ON key to exist graph . You have now in the stack the four initial values close enough to the root to use in the HP28C’s solveur (or any solveur of you alternative HP calculators).

|4:     (-6.4706,1.0000)|

|3:     (-1.6176,1.0000)|

|2:     (-0.8824,1.0000)|

|1:      (1.0294,1.0000)|

[SOLV][SOLVR]

 [  X  ] 

[shift][  X  ]

|#X:#1.0000#############|

|Zero                   |

|2:     (-0.8824,1.0000)|

|1:               1.0000|

  4  [shift][ ROLL ]

 [  X  ] 

 [shift] [  X  ] 

|#X:#-6.4495##############|

|Sign Reversal            |

|1:                -6.4495|

| X |EXPR=|   |   |   |   |

Process the same way for the two last estimates:

 4 [shift] [ ROLL ] [  X  ] [shift] [  X  ] 4 [shift] [ROLL][  X  ] [shift] [  X  ] 4 [shift] [ROLL] [CURSOR]

|4:              -6.4495|

|3:              -1.5505|

|2:              -1.0000|

|1:               1.0000|

For the second equation, process the same way and take advantage of your HP-28C to draw the curve. It is great helper in finding how to initiate root seeking. Making all previous HP a crabs

P.S.: For second equation { -3 -0.5 1.5 2.0 } obtained the same graphicaly way!

Edited: 28 June 2012, 5:10 p.m. after one or more responses were posted

The same straightforward syntax I used in my posting above.

Thanks for the programming syntax for the 28C. But what I actually needed was the syntax for the 42S to solve the quartic equation I referenced above.

The HP42S solver is acting exactly as one the HP28C does. Main differencies are that it only use a pair of initial values to start seeking. The HP28C may use a single, a pair or a triplet as initial(s) guess entered as a list { best_guess, low_interval_guess, high_interval_guess } .

On the HP42S, you first have to enter the equation as a program:

 [shift][PRGM]

     00 { 39-Byte Prgm }

           01>LBL "P_X"

           02 MVAR "X"

           03 4       

           04 RCL "X" 

           05 X^2     

           06 ×       

           07 31      

           08 -       

           09 RCL "X" 

           10 ×       

           11 21      

           12 +       

           13 RCL "X" 

           14 ×       

           15 18      

           16 +       

           17 RTN     

           18 .END.   

 [shift][PRGM]

Now you are ready to start the solver:

 [shift][SOLVER]

[  P_X  ]

 6 [+/-] [  X  ]  2 [+/-] [  X  ][ X ]

| X = -3                      |

[ X ][   ][   ][   ] [   ][   ]

 { -6 -4 } [  X  ][shift][  X  ]

Second root can be found using a different set of initial values:

 2 [+/-] [  X  ]  0 [  X  ][ X ]

| X =-0.5                    |

[ X ][   ][   ][   ] [   ][   ]

 0 [ X ] 1 [ X ] [ X ]

| X =1.5                      |

[ X ][   ][   ][   ] [   ][   ]

Finaly , last interval :

 1.8 [ X ] 5 [ X ] [ X ]

| X =2                        |

[ X ][   ][   ][   ] [   ][   ]

But, I sorry to repeat it again, it is hard to have efficient initial guess without a graph. And nothing is as easy and convenient as a graphic calculator to quickly plot it and zoom in regions of interest. Having the shape of the function is a great help to seek after its roots!

Thank you for your response. I thank everyone for their helpful responses. I totally agree with you about graphing. I have loved my 28C since I got it in 1986. As I said before, I have had a 45, 15C, 28C and a 50g. Of all, the 28C is my favorite. I'm glad I joined this forum. All of you have been incredibly helpful.

I've found the following program at Thomas Okken's website:

"G4_G3_G2.raw Solves 2nd, 3rd, and 4th order polynomials. using closed-form solutions. Contributed by Christian Vetterli."

 00 { 526-Byte Prgm }

 01>LBL "G2"

INPUT "A"  INPUT "B"

INPUT "C"  RCL "B"  X^2

RCL "A"  RCL× "C"  4  ×

-  SQRT  STO "C"

RCL "A"  2  ×  STO÷ "C"

+/-  STO÷ "B"  RCL "B"

RCL+ "C"  RTN  RCL "B"

RCL- "C"  RTN

 26>LBL "G3"

RECT  DEG  INPUT "A"

INPUT "B"  INPUT "C"

INPUT "D"  SF 00  CF 01

RCL "A"  STO÷ "B"

STO÷ "C"  STO÷ "D"

 39>LBL 15

RCL "B"  3  Y^X  13.5  ÷

RCL "B"  RCL× "C"  3  ÷

-  RCL+ "D"  STO "A"

RCL "B"  X^2  3  ÷

STO- "C"  RCL "C"  3  ÷

3  Y^X  RCL "A"  2  ÷

X^2  +  SQRT  RCL "A"  2

÷  X<>Y  X=Y?  +/-  X<>Y

-  3  1/X  Y^X  STO "D"

2E-3  STO 00

 82>LBL 16

FS? 01  GTO 01  3

STO÷ "C"  SF 01

 88>LBL 01

POLAR  1  120  COMPLEX

STO÷ "D"  RCL "C"  SF 25

RCL÷ "D"  RCL "D"  X<>Y

-  RECT  RCL "B"  3  ÷

-  RTN  ISG 00  GTO 16

 108>LBL "G4"

SF 00  CF 01  CF 02

RECT  DEG  INPUT "A"

INPUT "B"  INPUT "C"

INPUT "D"  INPUT "E"

RCL "A"  STO÷ "B"

STO÷ "C"  STO÷ "D"

STO÷ "E"  RCL "B"

STO "G"  X^2  RCL× "C"

16  ÷  RCL "B"  4  Y^X

3  ×  256  ÷  -  RCL "B"

RCL× "D"  4  ÷  -

STO+ "E"  RCL "B"  3

Y^X  8  ÷  RCL "B"

RCL× "C"  2  ÷  -

STO+ "D"  RCL "B"  X^2

0.375  ×  STO- "C"  2

RCL× "C"  STO "B"

RCL "C"  X^2  4

RCL× "E"  -  STO "C"

RCL "D"  STO "F"  X^2

+/-  STO "D"  XEQ 15

 175>LBL 05

STO ST Y  ABS  1E-8

FC? 02  X<=Y?  GTO 02

SF 02  ISG 00  XEQ 16

GTO 05

 186>LBL 02

RCL ST Z  SQRT  STO "A"

SF 25  STO÷ "F"  ISG 00

XEQ 16

 194>LBL 06

STO ST Y  ABS  1E-8

FC? 02  X<=Y?  GTO 02

SF 02  ISG 00  XEQ 16

GTO 06

 205>LBL 02

RCL ST Z  SQRT  STO "E"

SF 25  STO÷ "F"  RCL "E"

RCL+ "A"  RCL- "F"

XEQ 04  RCL "F"

RCL+ "A"  RCL- "E"

XEQ 04  RCL "E"

RCL+ "F"  RCL- "A"

XEQ 04  RCL "E"

RCL+ "F"  RCL+ "A"  +/-

 227>LBL 04

2  ÷  RCL "G"  4  ÷  -

STOP  RTN  .END.

XEQ G4

A?1 R/S

B?8 R/S

C?9 R/S

D?-8 R/S

E?-10 R/S

-1.55051025722 -i3.35403051129e-25

1 i6.25300999685e-25

-1 -i9.54592314951e-27

-6.44948974278 -i2.80352025407e-25

These match your results, considering the very small imaginary parts are actually zero.

Edited: 28 June 2012, 9:53 p.m.

Christian Vetterli's program above will compute all roots, real or complex, of polynomials up to degree 4 (if you don't mind keying in 200+ lines). Let it solve 4x^4-8x^3-13x^2-10x+22=0, for instance:

XEQ G4

A?4 R/S

B?-8 R/S

C?-13 R/S

D?-10 R/S

E?22 R/S    -->   3.11803398875 i0

     R/S    -->   -1 -i1

     R/S    -->   -1 i1

     R/S    -->   0.88196601125 i0

You can simplify the program above by deleting the three ENTER commands after RCL X. The HP-42S solver populates the stack with the independent variable, just as the HP-15C did. This is true for the integrator also.

Quote:

---

Two weeks ago my sons gave me an HP-42S

---

Thanks, we tried. One went to Notre Dame, the other University of Washington. Both had their minors in mathematics. I try to catch up! Unfortunately, their calculators of choice was the TI-89.

Hi.

Thanks for pointing that out, but as a matter of fact, those three ENTER are actually needed.

The HP42S does not actually work as the HP15C (*) because of one fact: you can have more than one unknown in an equation when writing them for the HP42S SOLVER so you can choose which unknown to solve for. That's why you may actually 'declare' the unknowns you want to solve for by using the 'MVAR' statement/function. Then your program simply returns the function value based upon the many unknowns you have previously declared. Because neither the user nor the calculator can predict which variable will be claimed for solving for, the program must be written having all possibilities being handled.

As you see, right after the 'RCL "X"' there is only one copy of the variable "X" in the stack register 'X'. If you do not copy it into the whole stack, the program will not compute the polynomial expression correctly because any previously existing values in Y, Z and T will be used, instead of the actual "X" value being tested. Think of an expression with two unknowns, like "A" AND "B", whether they repeat or not in the expression. You could choose to solve for either "A" or "B", once the other is known and previously defined (stored). Then the SOLVER will compute the same expression as many times as needed for the chosen variable, and its value will be updated in the variable name itself, not in the stack registers.

You see, when you have a program in the HP15C that is going to be used by the SOLVER or the numerical integrator, it must be programed in such a way that it returns the value of the function being evaluated based in one specific unknown. Some years ago the multivariable 'functionality' of the HP42S SOLVER was proposed for the HP15C by using the index register. It worked pretty fine, and the resulting programs are listed in the article forum somewhere in the far past...

Hopefully I expressed myself correctly... in English, at least.

Cheers.

Luiz (Brazil)

(*) it applies to the HP34C SOLVER and its implementation found in the MATH module for the HP41 as well.

Edited: 1 July 2012, 9:31 p.m.